Will it be always?

Number Theory Level 2

Let a be any positive integer, such that $2^a - 1$ is not divisible by 3 then $2^{a-1} - 1$ is divisible by 3 always.

True or False?

True False

1 solution

Atishay Patni
Nov 26, 2019

Let $2^a - 1$ is not equal to $3k$ where k is any positive integer. Now, $2^a$ will not be equal to $3k+1$ and we can simply also say that $2^a$ can never be equal to $3k$ . Therefore, we are only left with $3k + 2$ , so, $2^a = 3k + 2$ .

$2*(2^{a-1} - 1) = 3k$ .

I may have problems with english and have misinterpreted the wording ... but for me you showed that the statement is true , not false ?

joel Petitet - 1 year, 6 months ago

For a = 2, 2^2 - 1 = 3

José Antonio Fuentes - 1 year, 4 months ago

Of course 2^2-1=3 ... a cannot be equal to 2 if we want to have 2^a-1 not divisible by 3 In fact we can easily prove that if a is even , 2^a-1 is divisible by 3 , and if a is odd 2^a-1 is not divisible by 3 ( because 2^2=4=3*k+1 ) So : If 2^a-1 is not divisible by 3 then a is odd then a-1 is even then 2^(a-1)-1 is divisible by 3

( This is what Atishay Patni shows above with a different method ) The answer should be " True "

joel Petitet - 1 year, 4 months ago

Will it be always?

1 solution

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